Warmup for AP Calculus BC

Transcription

1 Nichols School Mathematics Department Summer Work Packet Warmup for AP Calculus BC Who should complete this packet? Students who have completed Advanced Functions or and will be taking AP Calculus BC in the fall of 08. Due Date: The first day of school How many of the problems should I do? ALL OF THEM How should I organize my work? You should show all work in a separate sheets of loose-leaf paper. If a problem requires a graph, then you should use graph paper. Keep your materials in a folder with -prongs. How will my teacher know that I ve done the work? Your teacher will collect your notebook on the first day of school. Your teacher may choose to QUIZ or TEST you on this material if he or she feels it is necessary BE PREPARED! How well should I know this material when I return? You should recognize that you ve seen this material before, and you should also be able to answer questions like the ones in this packet. If the material is revisited in your net class, it will only be for a brief amount of time your teacher will assume that all you need is a quick refresher. Note from your teachers: We feel that this summer work will truly help you succeed this year. We understand that summer is a time for relaation and fun, but it is imperative that you spend some time before you return reviewing your materials. This packet is mandatory, and you must treat it as you would any other etremely important homework assignment. You will be held accountable for this material. We also highly suggest that you do a bit of it at a time in the weeks leading up to school don t leave it for the last day!!!

2 Functions, Inverses, and Composition. A triangle is formed by the coordinate aes and a line through the point (,). a. Verify that y = +. b. Write the area of the triangle as a function of. Determine the domain of the function in the contet of the problem. c. Use a graphing calculator to determine the minimum area of the triangle Determine where the rational function R ( ) = is undefined. Decide whether there is a hole or an asymptote at such numbers. Make an accurate sketch of the graph.. A rectangle has one corner on the graph of y = 5, another at the origin, a third on the positive y-ais, and the fourth on the positive -ais. (see the pic below) a. Epress the area A of the rectangle as a function of. b. What is the domain of A ( )? c. Graph the function A ( ). d. Use a graphing calculator to determine the value of that gives the maimum area. 4. For the function f( ) = +, find the following: a. f () b. f( ) c. f( ) d. f( + h) 5. a. Graph the circle + y = 00 and the parabola + y = 00 b. Solve the system of equations: y = c. Where do the parabola and the circle intersect? y =. Warmup for AP Calculus BC Page

3 Functions, Inverses, and Composition 6. Use the graph of the function g shown: a. State the domain and range of g. b. Find g( ). c. List the & y intercepts. d. For what values of does g= ( )? e. Solve g> ( ) 0. f. Sketch the graph of y= g ( ). g. Sketch the graph of y= g ( ) +. h. Sketch the graph of y= g ( ). i. Sketch the graph of g. 7. Suppose Cgrepresents ( ) the cost C of manufacturing g cars. Eplain what represents. C (800,000) 8. The spread of oil leaking from a tanker is in the shape of a circle. If the radius r (in feet) of the spread after t hours is given by the function rt ( ) = 00 t, find the area A of the oil slick as a function of the time t. 9. If f = + and g ( ) = + a, find the value of a so that the graph of f ogcrosses the ( ) 5 y-ais at. 0. If the point (,-5) is on the graph of the function g, ( ) what point must be on the graph of g ( ), the inverse function?. For each rational function, identify: i) domain ii) the equations of any asymptotes iii) & y- intercepts. Use this information (along with any other helpful points) to create an accurate graph of the function. a. f( ) = 4 b. g ( ) = +. Give the equation of a rational function that has vertical asymptotes at = & = and also has a horizontal asymptote at y = 4. Warmup for AP Calculus BC Page

4 Limits, Functions, and Rates of Change. Find each limit using algebraic methods. Be sure to utilize proper notation. a. lim b. lim+ 6 c. limπ 4 + tan cos d. lim +. Use the accompanying graph of y = f( ). a. Find b. Find lim f( ) + lim f( ) c. Does eist? If so, what is it? If not, eplain why not. d. Determine whether f is continuous at each of the following numbers. If it is not, eplain why not. i) = ii) = iii) = iv) = 4. Determine the value of k that will make the function continuous at = 4 9, 4 f( ) = + k + 5, > 4 4. For the function f( ) = 4 : a. Find the derivative of f at =. b. Find the equation of the tangent line to the graph of f at the point (, 9). c. Sketch the graph of the function and the tangent line on the same set of aes. 5. The following table gives the distance s (in feet) that a parachutist has fallen over time t ( in seconds). Time, t a. Find the average speed from t = to t = 4 seconds. b. Find the average speed from t = to t = seconds. c. Find the average speed from t = to t = seconds. d. Use your graphing calculator to find the power function that best fits the data. (seconds) Distance, s (in feet) e. Using the function you found in part d, find the instantaneous rate of change at t = second. Warmup for AP Calculus BC Page

5 Additional topics: Sequences, Vectors, Trigonometry & Polar Coordinates. Determine whether the given sequence is arithmetic, geometric, or neither. Provide evidence for your conclusion. a. 6,,6,44,... b., 0, 8, 6,... n 8 c. + 7 d. 5,0,4,,... 5 e. Which of the sequences above, if epressed as an infinite series (add up the terms), has a finite sum?. Find the direction vector of each line: a. y = 4 b. 5 = 5. Write parametric equations for each of the lines in problem #5. 4. Solve each equation on the interval 0 θ < π a. 4sin θ = 0 c. sin ( θ ) + = 0 b. cos θ cosθ + = d. tanθ + = Write each polar equation in rectangular coordinates (,y). Identify the equation and graph it by hand. a. r = sinθ b. rcosθ + rsinθ = 6 6. Sketch the graph of the polar equation by hand; make a table of values ( θ, r) to generate and justify your graph. (A sheet of polar graph paper is included in this packet.) a. r = + cosθ b. r = sinθ 7. Find all values of θ on [0, π ) for which the following are true. a. cosθ = b. sinθ = c. cotθ = 0 Warmup for AP Calculus BC Page 4

6 Additional topics: Sequences, Vectors, Trigonometry & Polar Coordinates 8. Give the eact value of each of the following. π π a. sin b. cos c. sec(45 ) d. cot(5 ) e. 7π tan 6 9. For each angle on the unit circle shown, fill in the angles (in degrees and radians). You may write your work on this sheet. (Be sure to turn it in with your other work.) Warmup for AP Calculus BC Page 5

7 Derivatives. If f =, use the definition of the derivative to find f (). Note: you are finding ( ) 5 f () directly, not finding f ( ) and substituting.. If f( ) =, find f ( ) using the definition of the derivative. + Compute the derivative using appropriate short cuts, i.e. power rule, product rule, quotient rule, etc. y = ( ) 4. y = y = e ( ) sin 6. y = + 7. y = ln ( + ) 8. y= ln ( e) 9. If f( ) = ( ) 5, find f (0). 0. Given cos sin( ) y y y + =, find dy d. Warmup for AP Calculus BC Page 6